Dan Thompson

Conference, University of Warwick, July 2024

We extend Bowen's approach to thermodynamic formalism to flows on complete separable metric spaces. We define a suitable notion of specification in this setting, which gives uniform transition times for orbit segments which start and end in a fixed closed ball (with the transition time allowed to be larger if the ball is larger). The key point, particularly for the existence of an equilibrium state, is a Strong Positive Recurrence (SPR) assumption defined at this level of generality. As one application, we establish that for a sufficiently regular potential with SPR for the geodesic flow on a geometrically finite locally CAT(−1) space, there exists an ergodic Gibbs measure. This measure is finite, and is the unique equilibrium state. Joint work with Vaughn Climenhaga and Tianyu Wang.

Dynamics seminar, Northwestern University, November 2022

Consider a general CAT(−1) space and a bounded Hölder potential on the space of geodesics. We describe how to construct a Gibbs measure using appropriate weighted quasi-Patterson densities. If the Gibbs measure is finite, then it is an ergodic equilibrium state. We thus generalize results of Paulin, Pollicott, Schapira (for pinched negative curvature manifolds) and Roblin (for the 0-potential for CAT(−1) spaces). Unlike previous results in this direction in the CAT(−1) setting, our construction does not require a condition that the potential must agree over geodesics that share a common segment, which is a restrictive condition beyond the Riemannian case. The branching phenomenon is typical in CAT(−1) spaces and has been a major obstacle to fully developing the theory of equilibrium states in this setting. Joint work with Caleb Dilsavor (Ohio State).

AMS Sectional Meeting, Tufts, March 2022

We extend Bowen's specification-based results on uniqueness of equilibrium states to a wide class of non-compact systems. We define a suitable notion of specification in this setting, which gives uniform transition times for orbit segments which start and end in a compact set (with the transition time allowed to be larger if the compact set is larger). The key point is a Strong Positive Recurrence (SPR) assumption defined at this level of generality. As an application, we establish uniqueness of equilibrium states for SPR potentials for geodesic flow on geometrically finite CAT(−1) spaces.

Dynamics seminar, Brigham Young University, December 2018; Dynamics seminar, University of Houston, February 2019

Equilibrium states for geodesic flows over compact rank 1 manifolds and sufficiently regular potential functions were studied recently by Burns, Climenhaga, Fisher and myself. We showed that if the higher rank set does not carry full topological pressure then the equilibrium state is unique. In this talk, I describe new joint work with Ben Call, which shows that these equilibrium states have the Kolmogorov property. When the manifold has dimension at least 3 (for example, the Gromov example of a graph manifold) this is a new result even for the Knieper–Bowen–Margulis measure of maximal entropy.

Penn State Dynamical Systems Workshop, Pesin 70th Birthday Event, October 2017; Mathematical Congress of the Americas, Montreal (Canada), July 2017

The geodesic flow on a compact locally CAT(−1) metric space is a far-reaching generalization of the geodesic flow on a closed negative curvature Riemannian manifold. Our approach shows that such geodesic flows are Smale flows. A Smale flow is a topological flow equipped with a continuous bracket operation which is an abstraction of the local product structure from uniform hyperbolicity. By symbolic dynamics, we mean there exists a suspension flow over a shift of finite type which describes the original dynamics. By taking additional care in the construction, we verify that the roof function can be taken to be Lipschitz in our setting. With this additional ingredient, ergodic-theoretic results true for Axiom A flows are extended to this setting. For example, we obtain that the Bowen–Margulis measure for the geodesic flow is Bernoulli and satisfies the Central Limit Theorem. Joint work with Dave Constantine and Jean-François Lafont.

Geometry seminar, Yale, March 2016; Dynamics seminar, Michigan, February 2016

We establish results on uniqueness of equilibrium states for geodesic flows on rank one manifolds, applying machinery developed by Vaughn Climenhaga and myself which works when systems satisfy suitably weakened versions of expansivity and the specification property. The geodesic flow on a rank one manifold is a canonical example of a non-uniformly hyperbolic flow. Our methods are completely different from those used by Knieper in his seminal proof that there is a unique measure of maximal entropy in this setting. Joint work with Keith Burns (Northwestern), Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

British Mathematics Colloquium, Queen Mary's (London), April 2014; Pure Mathematics Colloquium, St. Andrews, April 2014

Generalised beta-transformations are the class of piecewise continuous interval maps given by taking the beta-transformation x ↦ βx (mod 1), where β > 1, and replacing some of the branches with branches of constant negative slope. We would like to describe the set of beta for which these maps can admit a Markov partition. We know that beta (the exponential of the entropy of the map) must be an algebraic number. Our main result is that the Galois conjugates of such beta have modulus less than 2, and the modulus is bounded away from 2 apart from the exceptional case of conjugates lying on the real line. This extends an analysis of Solomyak for the case of beta-transformations. I will also describe a connection with some results of Thurston's final paper, where the Galois conjugates of entropies of post-critically finite unimodal maps describe an intriguing fractal set.

Ergodic theory and dynamical systems seminar, University of Warwick, April 2014

We establish results on uniqueness of equilibrium states for the well-known Mañé examples of robustly transitive diffeomorphisms. This is an application of machinery developed by Vaughn Climenhaga and myself, which applies when systems satisfy suitably weakened versions of expansivity and the specification property. The Mañé examples are partially hyperbolic maps of the 3-torus. Joint work with Vaughn Climenhaga (Houston) and Todd Fisher (Brigham Young).

ICIAM conference, Ohio State MBI, May 2014

I describe an approach to coding sequence (CDS) density estimation in genomic analysis introduced by myself and David Koslicki. Our approach is based on the topological pressure, a measure of 'weighted information content' adapted from ergodic theory. We use the topological pressure (with suitable training data) to give ab initio predictions of CDS density on the genomes of Mus Musculus, Rhesus Macaque and Drosophila Melanogaster. While our method is not sufficiently precise to predict the exact locations of genes, we demonstrate reasonable estimates for the 'coarse scale' problem of predicting CDS density. Joint work with David Koslicki (Oregon State).

IUPUI, October 2013

Joint work with Vaughn Climenhaga and Kenichiro Yamamoto. We establish the large deviations principle for the family of S-gap shifts. Key to our approach is a 'horseshoe theorem' which allows us to approximate invariant measures by ergodic measures supported on sofic subshifts of the S-gap shift. Our large deviations result applies much more generally, based on machinery developed by Climenhaga and myself to prove uniqueness of equilibrium states in various different 'non-uniform' settings.

Midwest Dynamics Conference, October 2013

Based on a series of papers by Vaughn Climenhaga and myself developing a new approach to prove uniqueness of equilibrium measures for dynamical systems with various non-uniform structures. One class of model examples is the family of S-gap shifts — for a fixed subset S of the natural numbers, the corresponding S-gap shift is the collection of binary sequences where the length of every run of consecutive 0's is a member of S. The key difficulty is a spectacular failure of the Markov property. Our techniques can be adapted to apply to many other interesting dynamical systems beyond the uniformly hyperbolic case (e.g. beta-shifts, interval maps with parabolic fixed points, non-uniformly expanding maps in higher dimensions, some partially hyperbolic examples).

Mathematical Congress of the Americas, August 2013

For a broad class of symbolic dynamical systems without the Markov property, including the coding spaces of many piecewise continuous interval maps, we show how to approximate an arbitrary ergodic measure with a measure of almost the same entropy supported on a sofic subshift. This is interpreted as a symbolic analogue of a 'hyperbolic horseshoe' theorem. We present two ways to establish this result, both based on surgery on a single generic orbit — one based on Ornstein's d-bar metric, and the other on the theory of Kolmogorov complexity.

Conference on Thermodynamic Formalism, PUC (Chile), July 2013

An overview and report on recent progress for a long-term project joint with Vaughn Climenhaga concerning measures of maximal entropy and equilibrium states for a large class of dynamical systems with a 'non-uniform orbit structure', including piecewise continuous and parabolic interval maps, and some higher dimensional partially hyperbolic examples.

Maryland–Penn State Workshop on Dynamical Systems, Maryland, April 2011; Dynamics seminar, CUNY, May 2011

Joint work with Vaughn Climenhaga establishing uniqueness of equilibrium states for (1) a large class of shift spaces which includes every beta-shift, and (2) a large class of potential functions which strictly includes those with the Bowen property. As an application, our method yields new results in the theory of thermodynamic formalism for piecewise monotonic interval maps. Our method handles a variety of systems without a Markov structure, covering a class of potentials that are well behaved away from a 'small' set.

University of Washington, Seattle, April 2011

Based on joint work with Vaughn Climenhaga, in which we show that every shift space which is a factor of a beta-shift has a unique measure of maximal entropy. This provides an affirmative answer to Problem 28.2 of Mike Boyle's article 'Open problems in symbolic dynamics'. A measure of maximal entropy is a measure which witnesses the greatest possible complexity in the orbit structure of a topological dynamical system.

Logic Seminar, Pennsylvania State University, November 2010

For compact spaces, the theory of topological pressure and equilibrium states is a cornerstone of modern ergodic theory. I give a brief overview of various approaches to generalizing this theory to non-compact spaces, and present an elementary alternative definition of topological pressure in the non-compact setting. This definition turns out to be a generalization of the Kamae entropy. The definition assigns a non-negative number to each point in the space, interpreted as a sort of complexity — in Cantor space, this quantity is related to the Kolmogorov complexity.

Versions of this talk given at Penn State, Yale, Northwestern, Maryland and Rice

In 2004, Manning showed that the topological entropy of the geodesic flow for a surface of negative curvature decreases as the metric evolves under the normalised Ricci flow. I describe a curvature condition on the metric under which monotonicity of the topological entropy can be established for a short time. In particular, this criterion applies to metrics of negative sectional curvature which are in the same conformal class as a metric of constant negative sectional curvature.

Penn State, Working Geometry seminar, February 2010

The topological entropy is one of the key invariants in the theory of dynamical systems. When the dynamical system is a geodesic flow on a negatively curved manifold, the topological entropy has a very natural formulation as a geometric quantity — it is the exponential growth rate of volume in the universal cover. I'll sketch the classic proof of this due to Manning, then sketch Katok's classic proof that the topological entropy can be used to characterize which metrics on a surface are hyperbolic.

Penn State, Dynamical Systems seminar, October 2009; Boston University, Dynamical Systems seminar, October 2009

The beta-transformation f(x) = βx (mod 1) has been widely studied since its introduction by Rényi in 1957. We show that for the beta transformation, the set of points for which the Birkhoff average of a continuous function does not exist (the irregular set) is either empty or has full topological entropy and Hausdorff dimension. This result follows from a corresponding result about dynamical systems which satisfy a topological dynamical property we call almost specification — every beta-shift satisfies almost specification.

Seminar (part of themed semester in ergodic theory), University of Surrey, March 2009

It has long been thought desirable to generalise the standard theory of topological pressure and equilibrium states to non-compact spaces. Notably, Sarig developed the theory of Gurevic pressure for countable state shifts. In another direction, Bowen defined topological entropy for non-compact sets as a characteristic of dimension type. Pesin and Pitskel contributed a definition of topological pressure for non-compact sets generalising the Bowen definition. We give an elementary alternative definition of topological pressure in the non-compact setting via a suitable variational principle leading to an alternative definition of equilibrium state. We derive some properties of the new topological pressure and compare it with the other definitions.

London Mathematical Society one day ergodic theory meeting, University of Warwick, January 2009; Dynamical Systems Seminar, Northwestern University, October 2008

A recent weakening of the specification property provides new tools to study interesting systems beyond uniformly hyperbolic dynamics such as the beta-transformation. We call this property the almost specification property. We show that for dynamical systems with almost specification, the set of points for which the Birkhoff average of a continuous function does not exist is either empty or has full topological entropy. Every beta-shift satisfies almost specification, and we show that the irregular set for any beta-shift or beta-transformation is either empty or has full topological entropy and Hausdorff dimension.

Maryland–Penn State Workshop on Dynamical Systems, Penn State, October 2008

In 2004, Manning used an important formula of Katok, Knieper and Weiss to prove that as the metric on a negatively curved surface evolves under the (normalised) Ricci flow, the topological entropy of the geodesic flow decreases. In contrast, we observe that an example of Flaminio can be used to show that the Liouville entropy can either increase or decrease along a Ricci flow in a neighbourhood of a particular 3-manifold of constant negative curvature.

Maryland–Penn State Workshop on Dynamical Systems, University of Maryland, March 2008

We describe a result that applies to any dynamical system (X, T) with the specification property. Systems satisfying specification include any continuous map which is a factor of a topologically mixing shift of finite type and any topologically mixing continuous interval map. We show that, for a generic function f on X, the irregular set of f carries full topological pressure (in the sense of Pesin and Pitskel). Topological pressure is interpreted as a 'weighted' dynamical size, so the result says that the irregular set is as 'large' as it can be in an appropriate topological sense.

Dynamics Seminar, Warwick Mathematics Institute, January 2008; Dynamics Seminar, Manchester University, November 2007

In the 1980s, Pesin and Pitskel defined topological pressure for non-compact sets as a characteristic of dimension type, generalising Bowen's definition of topological entropy. We give an alternative definition of topological pressure in the non-compact setting via a suitable variational principle. We derive some properties of the new topological pressure and compare them with the properties of the Pesin and Pitskel pressure. We describe a simple example which illustrates the difference in the thermodynamic properties of the two quantities, and conclude with an example taken from the multifractal analysis of the Lyapunov exponent for the Manneville-Pomeau family of maps.

Conference 'Chaotic Properties of Dynamical Systems', Warwick Mathematics Institute, August 2007

The content was similar to 'The Irregular Set for Maps with the Specification Property Carries Full Topological Pressure' (but less general).

Postgraduate Seminar, Warwick Mathematics Institute, November 2006

A talk for a general postgraduate audience. We are interested in the average value of a function along the orbit of a point — the Birkhoff (or ergodic) average. It has been known since the 1930s that from the point of view of measure theory the Birkhoff average is very well behaved. However, from the topological viewpoint, this average behaves rather badly. Yet, by the so-called 'multifractal miracle', one can say a surprising amount about the destination of the Birkhoff sum. In the context of symbolic dynamics, I describe the 'multifractal miracle' in a way accessible to a general postgraduate audience.